cp's OEIS Frontend

A379667 Number of finite multisets of positive integers with sum + product = n.

%I A379667 #5 Jan 03 2025 23:29:39
%S A379667 0,1,1,1,2,2,3,3,5,5,6,7,8,8,11,12,13,14,15,17,19,19,20,22,26,26,29,
%T A379667 30,31,34,35,36,38,40,43,46,48,48,50,53,55,57,61,62,66,66,69,73,75,77,
%U A379667 79,82,83,85,89,91,94,94,95,103,106,107,111,113,116,119,121
%N A379667 Number of finite multisets of positive integers with sum + product = n.
%e A379667 The partition (2,2,1) has sum + product equal to 5 + 4 = 9, so is counted under a(9).
%e A379667 The a(0) = 0 through a(8) = 5 partitions:
%e A379667   .  ()  (1)  (11)  (2)    (21)    (3)      (31)      (4)
%e A379667                     (111)  (1111)  (211)    (2111)    (22)
%e A379667                                    (11111)  (111111)  (311)
%e A379667                                                       (21111)
%e A379667                                                       (1111111)
%t A379667 Table[Length[Select[Join@@Array[IntegerPartitions,n+1,0],Total[#]+Times@@#==n&]],{n,0,30}]
%Y A379667 Arrays counting multisets by sum and product:
%Y A379667 - partitions: A379666, antidiagonal sums A379667 (this)
%Y A379667 - partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
%Y A379667 - strict partitions: A379671, antidiagonal sums A379672
%Y A379667 - strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
%Y A379667 Counting and ranking multisets by comparing sum and product:
%Y A379667 - same: A001055 (strict A045778), ranks A301987
%Y A379667 - divisible: A057567, ranks A326155
%Y A379667 - divisor: A057568, ranks A326149, see A326156, A326172, A379733
%Y A379667 - greater: A096276 shifted right, ranks A325038
%Y A379667 - greater or equal: A096276, ranks A325044
%Y A379667 - less: A114324, ranks A325037, see A318029
%Y A379667 - less or equal: A319005, ranks A379721
%Y A379667 - different: A379736, ranks A379722, see A111133
%Y A379667 A000041 counts integer partitions, strict A000009.
%Y A379667 A025147 counts partitions into distinct parts > 1, non-strict A002865.
%Y A379667 A316439 counts factorizations by length, partitions A008284.
%Y A379667 A318950 counts factorizations by sum.
%Y A379667 A326622 counts factorizations with integer mean, strict A328966.
%Y A379667 Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A326152, A326178, A379720.
%K A379667 nonn
%O A379667 0,5
%A A379667 _Gus Wiseman_, Jan 03 2025